Orbit Calculations

Although teams need not compute orbits for Space Settlement Design
Competitions, it is important to recognize orbital mechanics as a
restriction on where things can go and stay in space. Some calculations
can illustrate this.

Orbits are ellipses, and the body being orbited (the Moon and most
satellites orbit the Earth; planets and interplanetary satellites orbit
the Sun) is at one of the foci. For the purposes of Space Settlement
Design Competitions, it is sufficient to generalize by considering only
circular orbits:

Use this equation to figure out how fast the Space Shuttle goes, or
how fast the Earth moves around the Sun. An exercise is included here
that demonstrates use of this equation for figuring out altitudes of
geosynchronous satellites.

An interesting feature of orbits is that the faster a satellite is
moving, the higher the altitude of its orbit. Because the distance
required to travel completely around the orbit grows directly as a
function of altitude (perimeter of a circle is pi x d), it takes longer to
complete a higher orbit than a lower one. This produces an odd phenomenon:
if the Space Shuttle follows the International Space Station in the
same orbit, the Space Shuttle must slow down and drop to a lower orbit
in order to catch up for a rendezvous with Space Station. Use the orbit
equation to figure out the relationship between orbital altitude and the
time required to complete an orbit.

Orbital mechanics also determines how long it takes to get from one
place to another. The most efficient way to get from one orbit (place)
to another in space is to perform a "Hohmann Transfer"; this is an
elliptical orbit that has one end of the ellipse on the orbit that is
being left, and the other end of the ellipse on the destination orbit,
as shown in the figure. The people who calculate orbits for a living are
most interested in the amounts of energy that will cause the satellite
to leave its original orbit on the proper trajectory, and will cause
it to stay in the new orbit. For Space Settlement Design Competitions,
we are most interested in the amount of time required to get from one
place (orbit) in space to another.

The time of flight for a Hohmann Transfer between two orbits is
defined by the following equation:

An exercise is included that shows how to use this equation to
calculate the time required to get to Mars. If we were to do this
calculation for other orbits, it would yield the following travel times
between possible destinations in the solar system:

Consider this table of transfer times from earth:

the Moon

5 days

Venus

146 days

Mars

259 days

the Asteroids

1.23 years (average)

Jupiter

2.74 years

Saturn

6.04 years

These mission durations can be reduced if more energy (fuel) is
expended to accomplish the orbital change. Expending less energy
will simply not get the satellite to the intended location, so these
represent maximum durations of direct missions to these destinations.

Getting to these destinations is complicated by the fact that
there are only certain times when you can go--you must select your
departure time so that you and the planet arrive at the same spot
on the planet's orbit simultaneously--if you choose another time,
you will get to the planet's orbit, but it won't be there. For the
Moon, this is merely a matter of selecting the appropriate time of
day. For the planets, however, mission planners may have to wait
years for Hohmann Transfer opportunities. For Space Settlement
Design Competitions, we do not expect teams to select actual mission
dates; we do, however, expect them to consider travel time to space
settlement locations in their schedules.

Where do communications Satellites Orbit?

Communication satellites are in geosynchronous orbit, which means
that they orbit around the Earth at the same rate the Earth spins on
its axis. They therefore appear to remain in the same spot over the
Earth.

This is the ONLY conventional circular orbit that is geosynchronous.
Think about what a satellite's orbit would look like from a point on
Earth if it were at this altitude, but inclined with respect to the
Equator. Remember, the orbit must be around the center of the Earth. (It
is not possible for a conventional spacecraft to stay above a single spot
that is not on the Equator. A design suggested by Dr. Robert Forward
would, however, enable a spacecraft to remain at a stationary location
above a different latitude through use of a solar sail; this "stacked
orbit" technique has not been implemented.)

How Long Does It Take to Get to Mars?

The time of flight for a Hohmann Transfer between two orbits is
defined by the following equation: